Monadic algebras of an involutive monoidal t-norm based logic

The main goal of this paper is to study universal and existential quantifiers on involutive monoidal t-norm based algebras, which are algebraic semantics for the logic of involutive left-continuous t-norms and their residua, and the resulting class of algebras will be called monadic IMTL-algebras. First we study some of their related algebraic properties and prove that the variety of monadic IMTL-algebras is the equivalent algebraic semantics of monadic predicate fuzzy logic $\mathbf{mMTL_{\forall}}$, which is equivalent to the modal fuzzy logic $\mathbf{S5(IMTL)}$, and show the completeness for $\mathbf{IMTL_{\forall}}$ via functional monadic IMTL-algebras. Moreover we start a systematic study of monadic algebraic structures that related to the monadic IMTL-algebras, some of which constitute the monadic MTL-algebras, monadic WNM-algebras, monadic NM-algebras, monadic BL-algebras, monadic MV-algebras and monadic Boolean algebras.Finally we give some representations of monadic IMTL-algebras. In particular, we character representable and directly indecomposable monadic IMTL-algebras by monadic filters.